Theorem lautcnv 40495

Description: The converse of a lattice automorphism is a lattice automorphism. (Contributed by NM, 10-May-2013.)

Hypothesis

Ref	Expression
lautcnv.i	⊢ 𝐼 = (LAut‘𝐾)

Assertion

Ref	Expression
lautcnv	⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → ◡𝐹 ∈ 𝐼)

Proof of Theorem lautcnv

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		lautcnv.i	. . . 4 ⊢ 𝐼 = (LAut‘𝐾)
3	1, 2	laut1o 40490	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
4		f1ocnv 6796	. . 3 ⊢ (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → ◡𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
5	3, 4	syl 17	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → ◡𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
6		eqid 2737	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
7	1, 6, 2	lautcnvle 40494	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ (◡𝐹‘𝑥)(le‘𝐾)(◡𝐹‘𝑦)))
8	7	ralrimivva 3181	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (◡𝐹‘𝑥)(le‘𝐾)(◡𝐹‘𝑦)))
9	1, 6, 2	islaut 40488	. . 3 ⊢ (𝐾 ∈ 𝑉 → (◡𝐹 ∈ 𝐼 ↔ (◡𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (◡𝐹‘𝑥)(le‘𝐾)(◡𝐹‘𝑦)))))
10	9	adantr 480	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → (◡𝐹 ∈ 𝐼 ↔ (◡𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (◡𝐹‘𝑥)(le‘𝐾)(◡𝐹‘𝑦)))))
11	5, 8, 10	mpbir2and 714	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → ◡𝐹 ∈ 𝐼)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052   class class class wbr 5100  ^◡ccnv 5633  –1-1-onto→wf1o 6501  ‘cfv 6502  Basecbs 17150  lecple 17198  LAutclaut 40390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-ov 7373  df-oprab 7374  df-mpo 7375  df-map 8779  df-laut 40394

This theorem is referenced by:  ldilcnv  40520